There are smooth manifolds, which are homeomorphic, but not diffeomorphic. It
happens to manifolds of dimension $>$3. This phenomenon
was discovered by
Milnor [1956] for the 7-dimensional sphere
$S^{7}$ in the fifties. A technique for classification of smooth
structures on a manifold of dimension $>$4 was developed
in the sixties. Homeomorphic, but not diffeomorphic manifolds were
found in all dimensions $>$4, see
Kervaire and Milnor [1963],
Barden [1965] and
Siebenmann [1971]. A substantial part of the technique used in these works is not applicable in dimension 4.

In the eighties Freedman [1982] gave a topological classification of closed
simply connected 4-manifolds and Donaldson [1987] proved that some smooth
closed simply connected 4-manifolds, which are homeomorphic
according to Freedman’s results, are not
diffeomorphic.

For smooth closed simply connected 4-manifolds, the only homotopy
invariant is the intersection form in the second homology with
integer coefficients: two manifolds of this kind are homotopy
equivalent iff their intersection forms are isomorphic (Whitehead
[1949], Pontryagin [1949]).

As
Freedman [1982] proved, for manifolds of this type,
homeomorphism is equivalent to homotopy equivalence. Thus, two
smooth closed simply connected 4-manifolds are homeomorphic iff
their intersection forms are isomorphic.

In contrast, on many closed simply connected
4-manifolds there are infinitely many smooth structures. Of course,
the number of smooth structures on a closed 4-manifold cannot be
uncountable (on a non-compact 4-manifold it can, as is the case for
$\mathbb{R}^{4}$). There is no conjectural classification scheme for
smooth structures on any closed 4-manifold.
Stern [2006] entitled
his survey paper by a question “Will we ever classify
simply-connected smooth 4-manifolds?”. In the introduction to his
paper he wrote: “The subject is rich in examples that demonstrate a
wide variety of disparate phenomena. Yet it is precisely this
richness which, at the time of these lectures, gives us little hope
to even conjecture a classification scheme.”

Proving that homeomorphic manifolds of dimension 4 are not diffeomorphic required
absolutely new tools. All the invariants, that have been used so far
for proving that some homeomorphic simply connected closed smooth
4-manifolds are not diffeomorphic, are deeply rooted in analysis.
They are based on counting solutions of some nonlinear partial
differential equations. Most of the results on non-existence of
diffeomorphisms between homeomorphic simply connected 4-manifolds
have been obtained using the Donaldson and Seiberg–Witten
invariants.

It is a long term challenge for topologists to find invariants which
would distinguish smooth structures on a 4-manifold, but would be
independent on analytic tools. This is not just an aesthetic issue:
the analytic tools are not convenient in some situation.

For example, in dimension 4 smooth structures are
closely related to PL-structures (piecewise linear structures). In
any dimension a smooth structure on a manifold determines on it a
specific PL-structure uniquely defined up to PL-homeomorphism. In
dimensions $\leq$6 any PL-structure can be obtained in this way from
a smooth structure and the smooth structure is unique up to
diffeomorphism, see Siebenmann [1971]. Hence any invariant of a smooth
structure on a 4-manifold depends only on the PL-structure. However,
in order to prove that two homeomorphic PL-manifolds of dimension 4
are not PL-homeomorphic, now one has to equip them with smooth
structures, and then calculate the invariants. The techniques for
calculation are not easy, especially if the smooth structure is not
defined naturally (say, as the underlying structure on a complex
surface), but just obtained by smoothing of a PL-structure.

It is worth mentioning a partial success of efforts
towards eliminating of analysis. An invariant of smooth 4-manifolds,
the Ozsváth and Szabó [2006] mixed invariant, conjecturally coinciding with the
Seiberg–Witten invariant, has been redefined (Manolescu et al. [2009]) in
purely combinatorial terms. However, on one hand, it is difficult to
calculate even for comparatively simple 4-manifolds, so it not an
effective tool for distinguishing smooth structures, on the other
hand, its combinatorial description has not clarified its nature,
has not related it to the rest of topology.

All the known invariants distinguishing smooth structures on closed
simply connected 4-manifold $X$ require non-trivial intersection
form on $H_{2}(X)$. In particular, for the 4-sphere these invariants
give nothing, they cannot help in disproving of the 4-dimensional
differential Poincaré conjecture, according to which any closed
4-dimensional manifold homeomorphic to $S^{4}$ is diffeomorphic to
$S^{4}$.

Thus, a development of new approaches to building invariants of
smooth structures is desirable.

One of the most powerful invariants, which distinguish smooth
structures on a 4-manifold, is the Seiberg–Witten invariant. It can
be presented in several forms. In particular, the Seiberg–Witten
invariant of a smooth closed simply-connected 4-manifold $X$ can be
identified with an element ${SW}_{X}$ of $\mathbb{Z}[H_{2}(X)]$, the integer
group algebra of the second homology group $H_{2}(X)$.

This identification makes the Seiberg–Witten invariant resembling
the Alexander polynomial of a link. For a classical link $L\subset S^{3}$, the Alexander polynomial is an
element of $\mathbb{Z}[H_{1}(S^{3}\smallsetminus L)]$, the integer group algebra of the
first homology group.

The Alexander polynomial can be defined in many ways. In particular,
it admits purely topological definitions.

The similarity between Seiberg–Witten invariant and Alexander
polynomial gives a hope to find either a definition of
Seiberg–Witten invariant free of heavy analysis, or, at least, to
invent similar invariants that solve the same problems, but are
defined and calculated in ways more traditional for topology. This
hope is supported by a relation between the Seiberg–Witten invariant
and the Alexander polynomial discovered by
Fintushel and Stern [1998].

Let $X$ be any simply connected closed smooth 4-manifold. Suppose
that $X$ contains a smoothly embedded torus $T$ with simply
connected complement $X\smallsetminus T$ and with zero self-intersection
number $T\circ T$. Since $T\circ T=0$, the normal bundle of $T$ is
trivial, and a tubular neighborhood of $T$ can be identified with
$T\times D^{2}$.

A knot surgery on $T$ takes away from $X$
the interior of a tubular neighborhood $T\times D^{2}$ of $T$ and
attaches $S^{1}\times(S^{3}\smallsetminus N_{K})$ instead, where $N_{K}$ is an open
tubular neighborhood of a smooth knot $K\subset S^{3}$. Observe that
the boundary of $S^{3}\smallsetminus N_{K}$ is diffeomorphic to the 2-torus
$S^{1}\times S^{1}$, and hence the boundary of the inserted piece
$S^{1}\times(S^{3}\smallsetminus N_{K})$ is diffeomorphic to the 3-torus
$S^{1}\times S^{1}\times S^{1}$ as well as the boundary $T\times\partial D^{2}$ of
the piece removed. The attaching is performed by a diffeomorphism
$S^{1}\times\partial(S^{3}\smallsetminus N_{K})\to T\times\partial D^{2}$ which maps
$\{pt\}\times l$, where $l$ is a longitude, i.e., a circle bounding
in $S^{3}\smallsetminus K$, to a fiber $\{pt\}\times\partial D^{2}$. This
requirement on the attaching map does not determine the map up to
diffeotopy, and hence does not necessarily determine $X_{K}$ up to
diffeomorphism. However, by the following theorem, under some
assumptions, all the manifolds obtained from the same $(X,T)$ and
$K\subset S^{3}$ by knot surgeries have the same Seiberg–Witten
invariant.

Theorem 1.

(Fintushel and Stern [1998]) Let $X$ be a simply connected closed smooth 4-manifold with
$b_{+}(X)>1$, let $X$ contain a smoothly embedded torus $T$ with
$T\circ T=0$ and simply connected complement $X\smallsetminus T$. Let $K$
be a knot in $S^{3}$. Then the result $X_{K}$ of a knot surgery is
homeomorphic to $X$ and

${SW}_{X_{K}}={SW}_{X}\cdot\Delta_{K}(t^{2}),$

where $t\in H_{2}(X)$ is the homology class realized by $T$ and
$\Delta_{K}$ is the symmetrized Alexander polynomial of $K$.

Present definitions of the Seiberg–Witten invariant are not
applicable to $(S^{3}\smallsetminus K)\times S^{1}$ or $(S^{3}\smallsetminus N_{K})\times S^{1}$. So, one cannot speak about ${SW}_{(S^{3}\smallsetminus K)\times S^{1}}$.
However, if the ${SW}$ was extended to this setup and satisfied a
reasonable additivity property, Fintushel–Stern Theorem would
suggests that ${SW}_{(S^{3}\smallsetminus K)\times S^{1}}$ should be
$\Delta_{K}$.

The Alexander polynomial of a classical knot $K$ is the order of
$\mathbb{Z}[H_{1}(S^{3}\smallsetminus K)]$-module $H_{1}(\widetilde{S^{3}\smallsetminus K})$,
where $\widetilde{S^{3}\smallsetminus K}\to S^{3}\smallsetminus K$ is the infinite
cyclic covering. The automorphism group of this covering is
$H_{1}(S^{3}\smallsetminus K)=\mathbb{Z}$, it acts in the homology group
$H_{1}(\widetilde{S^{3}\smallsetminus K})$, turning it into a
$\mathbb{Z}[\mathbb{Z}]=\mathbb{Z}[t,t^{-1}]$-module.

This module is called the Alexander module of $K$. It is
finitely generated and admits a square matrix of relations. The
determinant of this matrix is the Alexander polynomial. It is
defined up to multiplication by units of $\mathbb{Z}[\mathbb{Z}]$ (that is by
monomials $\pm t^{k}$).

According to this construction, the Alexander polynomial happens to
belong to $\mathbb{Z}[H_{1}(S^{3}\smallsetminus K)]$. The Seiberg–Witten invariant of
$X$ belongs to $\mathbb{Z}[H_{2}(X)]$. What could be a space $Y$ such that
$H_{1}(Y)=H_{2}(X)$?

There is an obvious candidate for such $Y$, the loop space $\Omega X$ of $X$. Indeed, $\pi_{i}(\Omega X)=\pi_{i+1}(X)$, and, in
particular, $\pi_{1}(\Omega X)=\pi_{2}(X)$; in the case of simply
connected $X$, $\pi_{2}(X)$ is isomorphic to $H_{2}(X)$ by the Hurewicz
theorem. Thus $\pi_{1}(\Omega X)=H_{2}(X)$. Therefore $\pi_{1}(\Omega X)$
is commutative, and hence $H_{1}(\Omega X)=\pi_{1}(\Omega X)=H_{2}(X)$.

First, let us make it closer to the smooth structure of $X$. The
loop space $\Omega X$ contains the space $\Omega_{Diff}X$ of
differentiable loops $S^{1}\to X$ that have at the base point fixed
non-vanishing differential. The replacement of $\Omega X$ by
$\Omega_{Diff}X$ does not change the homotopy type: well-known
approximation theorems imply that $\Omega_{Diff}X$ is a deformation
retract of $\Omega X$.

Let $KX\subset\Omega_{Diff}X$ be the subspace which consists of
loops that are smooth embeddings. Denote $\Omega_{Diff}X\smallsetminus KX$
by $DX$. Observe that $\operatorname{codim}_{\Omega_{Diff}X}DX=2$, hence the
inclusion homomorphism $\pi_{1}(KX)\to\pi_{1}(\Omega_{Diff}X)=\pi_{1}(\Omega X)$ is onto.

The action of $H_{1}(\Omega X)$ in
$H_{*}(\widetilde{\Omega X})$ is trivial, since $\Omega X$ is an
$H$-space, while $KX$ is not an $H$-space and
$H_{*}(\widetilde{KX})$ may be an interesting
$\mathbb{Z}[H_{2}(X)]$-module. It has a broad range of invariants
belonging to $\mathbb{Z}[H_{2}(X)]$ similar to the Alexander polynomial.
Indeed, if $X$ is simply connected, then $H_{2}(X)$ is a free abelian
group of finite rank $r$, and $\mathbb{Z}[H_{2}(X)]$ is isomorphic to the ring
$\Lambda_{r}=\mathbb{Z}[t_{1},t_{1}^{-1},t_{2},t_{2}^{-1},\dots t_{r},t_{r}^{-1}]$ of
Laurent polynomial in $r$ variables with integer coefficients, the
same ring as in the situation of the Alexander module of a classical
link.

A finitely generated module $M$ over $\Lambda_{r}$ gives rise to a
filtration of $\mathbb{Z}[H_{2}(X)]$

by Fitting ideals. The $i$th Fitting ideal $\operatorname{Fitt}_{i}(M)$ is
generated by the minors (determinants of submatrices) of order $r-i$
of the matrix of defining relations for $M$ In the topological
literature Fitting ideals are called also elementary ideals.
A generator of the minimal principal ideal containing $\operatorname{Fitt}_{i}(M)$
is denoted by $\Delta_{i}(M)$.

In the context of link theory, when $\Lambda_{r}=H_{1}(S^{3}\smallsetminus L)$,
where $L$ is an $r$-component link and $M=H_{1}(\widetilde{S^{3}\smallsetminus L})$ where $S^{3}\smallsetminus L$ is the maximal abelian covering space of $S^{3}\smallsetminus L$, the Laurent polynomial $\Delta_{i}(M)$ is called the $i$th
Alexander polynomial of $L$. The 0th Alexander polynomial is one
of the oldest link invariants. It was introduced by Alexander [1928].

Similarly, for a smooth simply connected closed 4-manifold $X$, each
$H_{i}(\widetilde{KX})$, as a module over $\mathbb{Z}[H_{2}(X)]$, gives rise to
a sequence of Laurent polynomials. They resemble the Seiberg–Witten
invariant. At least, they belong to the same $\mathbb{Z}[H_{2}(X)]$. The
modules $H_{i}(\widetilde{KX})$ are also invariants of $X$.

The problem:Does there exist homeomorphic
smooth simply connected closed 4-manifolds $X_{1}$, $X_{2}$ such
that $H_{*}(\widetilde{KX_{1}})$ and $H_{*}(\widetilde{KX_{2}})$
are not isomorphic?

The author discussed the problem with many leading specialists in
the field and asked them this question. The answers spread over a
broad spectrum. Only one expert expressed a definitely negative
opinion. On the other end, there were also very enthusiastic
reactions.

The most convincing argument is that for different smooth structures
on the same topological manifold $X$ a topologically observable
difference between smooth structures was the minimal genus of a
smoothly embedded surface realizing an element of $H_{2}(X)$. For any
smooth structure each class can be realized by an immersed sphere
with transverse self-intersection (by the Hurewicz theorem and
transversality), but the minimal number of double point of such
immersion depends on the structure. It is greater than or equal to
the minimal genus of a smoothly embedded surface realizing the
class.

An immersion $f:S^{2}\to X$ can be used to produce a loop in the space
of loops $\Omega_{Diff}X$. A loop in $\Omega_{Diff}S^{2}$ sweeping
the whole $S^{2}$ composed with $f$ gives a loop in $\Omega_{Diff}X$.
Generically this gives a loop in $KX$, because under a generic
choice of loop in $\Omega_{Diff}S^{2}$, points in the preimage of a
double point of $f$ are passed at different moments. However, in
families of loops realizing elements of high-dimensional homology
groups $H_{k}(\Omega_{Diff}S^{2})=\mathbb{Z}$ these pairs of points appear on
some of the loops. Such a family does not realize a homology class
in $KX$ or $\widetilde{KX}$.

How to calculate $H_{i}(\widetilde{KX})$? One may apply Vassiliev’s (1994)
idea, which led to discovery of the Vassiliev knot
invariants: start the calculation with a study of the dual
cohomology, the cohomology of the space of singular
knots. The space of singular knots has a rich geometric structure.

The space $DX$ consists of differentiable loops that are not
embeddings. It fits to the collection of discriminant hypersurfaces
studied by Vassiliev [1994]. The universal covering
$\widetilde{\Omega X}\to\Omega X$ defines a covering
$\widetilde{DX}\to DX$.

Resolve singularities of the discriminant $DX$, as Vassiliev did.
This gives rise to a filtration in $H^{*}(\widetilde{DX})$. The first
terms of these filtration are easier to calculate than the whole
cohomology group.

At first glance, the situation is much more complicated than in the
original setup in the theory of Vassiliev knot invariants. Let us
examine the extra difficulties.

First, instead of the space of singular knots, we have to deal with
its covering space. What does happen, when we pass from knots to
points of a covering space?

In general, if $p:X\to B$ is a covering, $b_{0}\in B$, $x_{0}\in X$ are
points such that $p(x_{0})=b_{0}$ and $X$ is path connected, then a
point $x\in X$ is uniquely determined by its image $p(x)\in B$, a
path $s:I\to B$ such that $s(0)=b_{0}$, and its covering path
$\widetilde{s}:I\to X$ starts at $x_{0}$ and finishes at $x$. The path
$s$ is defined up to path homotopy and multiplying by loops covered
by loops in $X$.

In particular, a point of $\widetilde{\Omega X}$ (no matter if it
belongs to $\widetilde{KX}$ or $\widetilde{DX}$) is defined by a
loop $u:S^{1}\to X$ and a continuous map $f:D^{2}\to X$ with
$f|_{S^{1}}=u$ considered up to homotopy which is fixed on $S^{1}$. The
action of $H_{2}(X)$ in $\widetilde{\Omega X}$ is realized in this
model as addition to the homotopy class of $f$ homotopy classes of
maps $S^{2}\to X$ realizing elements of $H_{2}(X)$.

Second, the space of all loops in the Vassiliev setup is
contractible (Recall that loops there have the base point and the
tangent vector at the base point fixed). This allowed to apply the
Alexander duality between the homology of the discriminant and
cohomology of its complement, the space of knots. This was done in a
finite dimensional (say, polynomial) approximation of the loop
space.

Here the loop space is not contractible. Therefore, the Alexander
duality is not applicable. However, it may be replaced by the
Alexander–Poincaré duality between the homology of
$\widetilde{KX}$ and a relative cohomology, the cohomology of
$(\widetilde{\Omega_{Diff}X},\widetilde{DX})$.

The homology of loop space $\Omega_{Diff}X$ and its universal
covering accommodate a rich structure of the string topology
operations (Chas and Sullivan [1999]). Geometrically, one can expect that this
structure incorporates the same information as the natural
filtration of the discriminant and the Alexander–Poincaré duality.
Apparently the connection have not been investigated. Nonetheless,
it would be natural to expect that the string topology is as strong
as the invariants discussed above.

Problem.Is the string topology sensitive to
smooth structures on 4-manifolds?

It would be interesting also to find direct relations between the
Vassiliev invariants theory and the string topology in the setup of
classical knot theory.